The spectral radius of edge chromatic critical graphs

A connected graph G with maximum degree Δ and edge chromatic number χ′(G)=Δ+1 is called Δ-critical if χ′(G−e)=Δ for every edge e of G. In this paper, we consider two weaker versions of Vizing's conjecture, which concern the spectral radius ρ(G) and the signless Laplacian spectral radius μ(G) of G. We obtain some lower bounds for ρ(G) and μ(G), and present some cases where the conjectures are true. Finally, several open problems are also proposed.